Emergent constraints on carbon budgets as a function of global warming

Earth System Models (ESMs) continue to diagnose a wide range of carbon budgets for each level of global warming. Here, we present emergent constraints on the carbon budget as a function of global warming, which combine the available ESM historical simulations and future projections for a range of scenarios, with observational estimates of global warming and anthropogenic CO2 emissions to the present day. We estimate mean and likely ranges for cumulative carbon budgets for the Paris targets of 1.5 °C and 2 °C of global warming of 812 [691, 933] PgC and 1048 [881, 1216] PgC, which are more than 10% larger than the ensemble mean values from the CMIP6 models. The linearity between cumulative emissions and global warming is found to be maintained at least until 4 °C, and is consistent with an effective Transient Climate Response to Emissions (eTCRE) of 2.1 [1.8, 2.6] °C/1000PgC, from a global warming of 1.2 °C onwards.

consequence, the slope esfimates obtained from Figure 2a are biased.There are papers that deal with this, the original one due to Bowman et al. (2018) who present a solufion using a hierarchical stafisfical modeling framework.Hierarchical emergent constraint (HEC) modeling accounts for error in the xvariable and yields accurate and precise esfimates, unlike those in Figure 2b.This framework has been applied in a number of arficles; see, for example, Thackeray and Hall (2019), Barkhordarian et al. (2021), Shiogama et al. (2022), and Chen et al. (2022).The submifted manuscript needs to fit the emergent relafionship based on HEC because the x-variable comes with uncertainty.
We give more specific comments, by secfion, below.

Introducfion
• While the introducfion is about remaining carbon budgets, most of the rest of the manuscript concentrates on the cumulafive carbon budget (see conclusions and Table 1).The discussion around Figure 4c on lines 161-166 shows what remains as a funcfion of global warming, and we suggest those results also be included in Table 1.The 30 PgC difference from IPCC6 (lines 179-180) is not significant.An emergent-constraint analysis needs to recognize properly the uncertainty in both y and x in order to provide policy-makers with accurate (unbiased) and precise (smallest uncertainty) results.

Method
• The proposed method consists of several components: the modeled carbon budgets, a linear, emergent relafionship, and the observafional constraint.While discussion for the first one is well presented, those for the other two are quite limited.The linear emergent relafionship deserves at least an equafion, and it should be shown clearly how it is used to obtain the emergent constraint along with its uncertainty.• Please provide a clear definifion of the observafional constraint.Throughout the paper, it is either presented as a central esfimate +/-errors, or a probability density funcfion of the predictor in the Appendix.Somefimes the error bounds are not symmetric (e.g., Figure 4c); however, the +/− results given in the text assume they are symmetric.

Results
• In Bayesian stafisfics, the prior and posterior are usually distribufions on the same variable, the former incorporafing prior informafion and the lafter being an update given new informafion.In fact, the stafisfical model in this manuscript is on the bivariate quanfifies (x, y), where P(x, y) = P(y | x)P(x), and interest is in the marginal distribufion, P(y) = R P(x, y)dx.This is not an analogue of Bayes theorem (line 356).The theorem involves the inverse probability P(x | y), which is not of great interest here for policy-making.The authors use "prior" for P(x) in Figure 3, which will cause confusion for readers.
• Why did the authors use median instead of mean for the results based on emergent constraints?The uncertainty calculafions for medians is different than that for means; they cannot be used interchangeably, since the sample median has a different standard error than that of the sample mean.

Conclusions
• The remaining carbon budget will be "spent" over fime.In the 30 years between 2020 and 2050 (for example), what is the target spend-rate for 1.5 •C, 2.0 •C, and 2.5 •C?How does that compare to the spend-rates up to 2020, where uncertainfies should be incorporated into the comparison?These might be quesfions to address using the emergent-constraint analysis.

Appendix
• Uncertainty calculafion for specific carbon budget (line 346).Using an addifive formula to calculate uncertainty is suitable when one wants to compute var(U + V ).However, it is not true that This is in effect what the authors are proposing on line 346; one has to be careful when combining relafive uncertainfies.
• The formulas for the uncertainty σf (x) on lines 365-374 assume that x is known with certainty.
However, x has uncertainty, and the authors esfimate the uncertainty on y taking into account the uncertainty on x, as specified on lines 360-363.What then is the relafionship between what is presented on lines 360-363 and lines 365-374?The two derivafions do not result in the same uncertainty measure.This is another example of the need to use HEC from the beginning, where the emergent-relafionship model has uncertainty in x, and every formula takes that into account.
• Uncertainty calculafion for the remaining carbon budget.Nofice that remain = total−hist or r = y − x.
The authors claim that r and x are independent, which is almost certainly not true; since the total y is constrained for a given level of global warming, r and x will be negafively correlated.The appropriate calculafion is where σ 2 y and σ 2 x are the variances of y and x, and ρxy is the correlafion between x and y.In the manuscript, ϵ 2 total and ϵ 2 hist are esfimates of σ 2 y and σ 2 x, respecfively.The authors sfill need to esfimate the third term.
• Line 168: "K" → •C (and throughout the manuscript) • Line 301: "Timeseries" → Time series • Lines 319, 328: "annual mean, global mean" → annual mean, global total?(Note: The unit of flux is "PgC/fime/area".To obtain the unit of "PgC/fime" that the authors use, a total over "area" is needed.This applies to global, land, ocean, or any region of interest.

3-2
Not clear what you mean by "specific" carbon budget vs just a carbon budget.Please clarify in text.

Response:
In the submitted manuscript we wrote: "We look for an emergent relationship between the specific carbon budget (cumulative emissions per unit of global warming) up to the current day and the specific carbon budget at each level of global warming.Perfect linear relationships passing through the origin would lead to an invariant specific carbon budget as a a function of global warming".In retrospect that may have been too cryptic, so we have replaced this text with: "We do not attempt to fit the slopes shown in Figure 2a, but instead seek an emergent relationship between cumulative emissions and global warming up to to the end of 2020, and the cumulative emissions at each future level of global warming".

We have also added this text to the Methods section: "We look for emergent relationships between the cumulative emissions calculated from equation 2, and global warming. As we want to constrain the policy-relevant carbon budgets for the Paris climate targets, we work in terms of the specific carbon budget (i.e. the cumulative emissions per unit of global warming). This allows us to combine model differences and observational uncertainties in both global warming and cumulative emissions, into a single metric (see 'Observational Constraint' below)".
3-3 It's not real clear what your point is here.Specifically what is meant by "invariant".If the linearity argument were strictly true, then given no change in emissions one would have no change in temperatures-so no offset.That is not the case, which is not a problem per se if one is only concerned with the increase.However, the slopes are not the same depending on the scenarios because they cover increasingly larger ranges in temperatures.That's pretty clear from the NorESM models.As seen in Figure 2b, the slopes (PgC/C) are pretty different depending on the scenario, which is more of a statement that problem is non-linear, not linear since for higher scenarios, more carbon is getting added in.
Response: This text has been rewritten in response to your other comments and no longer contains "invariant" (see response to 3-2).
3-4 I don't think it is appropriate to put multiple scenarios together.As can be seen from Fig. 2a 7-4 I don't think this is correct.You're calculating the ratio Z PgC/C = (CumEmis+noise1)/(Tanomaly + noise2) The resulting pdf is not a Gaussian with standard deviation that is the sum of the quadrature of noise1 and noise2.In other words, the quotient of two Gaussians is not a Gaussian.Take a look at Simon, M. ( 2002).Probability distribution involving Gaussian random variables.Boston: Kluwer Acad.Publishers.You can estimate it using a Monte Carlo, but there does exist (I believe) an analytic solution for the first and second moments.
Response: To be frank, there is also no good reason to assume that the noise in the cumulative emissions or in the decadal mean temperature anomaly are Gaussians.So to us, it would seem to be spuriously accurate to assume that they are Gaussian and to use that assumption to derive an 'exact' non-Gaussian distribution for their ratio.Instead we have included a caveat: "In the absence of detailed information on the distributions of the random errors in the cumulative emissions and the decadal mean temperature anomalies, we assume that the uncertainty in the observed specific carbon budget is Gaussian distributed".
8-1 This is not quite correct.Per Fig. 2, y is the cumulative emissions at 2C and x is the cumulative emission per change in temperature *of the models*, not the observations.There needs to be another conditional distribution that accounts for the PgC/C model distribution *given* observations of that distribution.Right now, the P(y|x) doesn't account for the correlation between the x and y of the implicit joint Gaussian distribution.In other words, P(y|x) is the same whether you have r^2=1 or r^2=0.1.I appreciate that approach has been used in the past, but there have been a number of publications since then that explicitly account for r^2 and the observational uncertainty, e.g., Sansom et al, 2021, Bretherton and Caldwell, 2020, Bowman et al, 2018; Please engage with those approaches and provide results on the dependency of these results to r^2 and the obs.error.
Response: We may have been unclear about the nature of our method here, so we have clarified by adding the following text: "The emergent constraint P(y) is therefore affected by both the quality of the emergent relationship P(y|x) and the uncertainty in the obsetvational constraint P(x)".
Our method has previously been compared with more sophisticated but less transparent Bayesian approaches (e.g.Samson et al., 2021) and has been found to give very similar results when the emergent relationship is strong (Nijsse et al., 2020), as it is here.For example the PDF on the specific carbon budget for 2 o C of global warming that we derive from ordinary-least-squares is very similar to the PDF that we derive from a a hierarchical Bayesian approach: We therefore prefer to stick with the more transparently simple OLS approach, although we acknowledge the reviewer's point by adding the following text:

"Our statistical approach follows many other previous emergent constraint studies in applying an ordinary-least-squares (OLS) fit between the predictand and the predictor variable. More sophisticated Bayesian approaches have been proposed and applied in other studies ( Bowman et al., 2018; Bretherton and Caldwell, 2020; Samson et al., 2021). These approaches yield very similar constraints to OLS when the emergent relationships are strong and linear (Nijsse et al., 2020), as they are here."
8-2 There's every reason to believe that they would be correlated.You can say that you have to assume this because you don't have a better solution, but the assumption itself is questionable.
Response: this is a fair comment, and one that really got us thinking.The correlation ρxy between x and y is of course encoded in an emergent relationship across models.Therefore, to avoid the possibility of making inconsistent assumptions concerning the correlation between the historical carbon budget and the total carbon budget, we have instead looked for a direct emergent constraint relating remaining carbon budgets for each level of global warming and the specific carbon budget up to he current day.The results of this approach are summarised in the new Figure 5.

11-1
This figure needs to be reworked.I can't see the lower SSPs, they are all on top of each other from 1-2C and the lower ones don't extend past out to the higher temperature anomalies.

Response:
The SSPs are all on top of each other because the lines are largely independent of scenario (which is a very useful thing, see response to 2-8).The lower scenarios don't extend out to the higher temperatures because global warming doesn't get that far for the lower scenarios.
11-2 How does the pink range change as a function of correlation and observational noise?Response: As the emergent relationship has a high r 2 , the uncertainty in the emergent constraint (i.e. the width of the pink range) varies approximately proportionally to to the assumed uncertainty in the observational constraint.
11-3 Need a similar plot (in supplemental) of the Gaussian prior and observations of the historic emissions per unit warming.

Response:
The Gaussian prior for the specific carbon budget of the model ensemble is shown by the black line in Figure 3a.The observations of the historic emissions per unit of global warming are shown by the black stars in Figure 2a.We don't feel that we need to repeat this information in a separate plot.

Figure 2b includes all of our SSP scenarios as we aim to find 4-1
, one would get different slopes (PgC/C) for each ESM ensemble for a given scenario.Consequently, the emergent constraint relative to a 2C change would be a function of scenario.Looking at Fig 2b, more closely if the ensembles were regressed on scenario, which is the right thing to do, then the slopes (and r^2) would be quite different.You need to justify why the scenarios should be thrown together (since they are correlated with each other).Why should we be looking at SSP585, which goes all the way pass 4C, when you're only interested in 2C?If that slope is different (among ESMs) from the same calculation at SSP370, what is that telling us about the ESM TCRE?I'd suggest choosing one scenario (closest to 2C) and putting the other scenarios in supplemental.The fact that regardless of scenario, 300-700 PgC/C can't all be correct-and observations can help through EC, is the main point from my perspective.As a matter of craft, the conclusions is pretty weak and doesn't really attempt to engage with how this work fits in the broader context.It doesn't really acknowledge any weaknesses, or how the approach could be improved in the future, or any recommendations/implications for policy/science.Needs to be beefed up.I don't see the justification for calculating a temperature anomaly over just the last decade.The argument is based upon the historic temperature anomaly relative to historic cumulative emissions, not just the last decade.Need to see how sensitive the temperature anomaly calculation is to the time range.Where does 0.12 C come from?To first order, I would expect it to be related to the distribution of temperature measurements.But, this needs to be clarified.
utility of carbon budgets to guide policy is hugely diminished.In fact, the scenario sensitivity of our emergent constraints is weak, as we now show by calculating emergent constraints for each SSP separately.We include a new table in the supplementary material which summarizes those results, which we refer to from the main text: "Response: We have added text and a reference to clarify this point: "

The figure of 0.12 o C comes from IPCC AR6 WG1, Chapter 2 (Cross Chapter Box 2.3, Table 1, Footnote b) that states a 'likely uncertainty range of +/-0.12 o C' for the decadal mean global warming relative to the 1850-1900".
7-3 I dont see how that is getting calculated.You can't just cite the whole IPCC.Provide a figure reference at least.Is this cumulative fossil fuel emissions or net emissions?Response: To clarify, we have added: "